Peter van Inwagen and Jonathan Bennett developed a simple and influential argument that the principle of sufficient reason (PSR) entails that there is no contingency in the world. Everything that happens, necessarily happens. The problem is deeper than it first appears. If PSR is true and God explains everything else, as many theists believe, then the cost of preserving God’s perfect rationality is the loss of His freedom, His moral perfection, and His providence and sovereignty. We are left with a single necessary world. The costs also include the loss of contingency and moral agency among created beings in the world. On the other hand, the cost of abandoning His perfect rationality is the unintelligibility of the world (which has disastrous effects on our reasoning with one another) and the unintelligibility of God’s actions in it. We cannot easily give up PSR.

The argument can look gimmicky, and I’ve heard that criticism of it. But it’s no gimmick. Suppose we talk of the explanation of states of affairs derivatively, and the explanation of propositions directly. For every true contingent proposition p, PSR requires that there is some explanation q such that q explains p if and only if ☐(q ⊃ p). But the conjunction P of all true contingent propositions is a contingent proposition, so P too has an explanation. Let G be the explanation of P. G must be a necessarily true proposition, since otherwise G would be a conjunct in P. But then G would be a contingent proposition that explained, among other things, itself. No contingent propositions explain themselves. So, P has an explanation just in case for some Q, ☐(Q ⊃ P) & ☐Q. But it follows uncontroversially from this that ☐P, and we are left with all of the unwanted implications above. I think there’s a neat way out of this problem that does not require weakening PSR, or adopting a weaker notion of explanation or defending infinite chains of contingent explanation.

Make the observation that if the conjunction P is contingent and true, then so is the conjunction P & ◊P’, where P’ is itself a conjunction of all true propositions in possible world W’. (◊P’ is just the proposition that possibly P’). Let the larger conjunction be R. Now, we know that R is contingent, but we also know that R is necessarilycontingent. There is no possible world in which the conjunction of propositions in R are all true and there is some necessary Q that explains R. That is, there is no world in which there is some Q such that ☐(Q ⊃ R) & ☐Q. So, it is impossible to derive ☐R from any necessarily true proposition. It is not merely the case that R is a contingent proposition with no explanation; rather, it is the case that R is a contingent proposition that is impossible to explain. But surely PSR cannot entail that a perfectly rational being would explain contingent propositions that are impossible to explain. PSR requires that every proposition that could be explained should have an explanation. Since there are lots of contingent propositions like R, we can have our cake and eat it too. We have every proposition explained that can be explained, and we have contingency in the world. And we avoid all of the unwanted implications of the loss of contingency noted above.

I think that this proposal poses a number of problems for theism, however. Notice that the solution was to pose the existence of a proposition R which was brute & contingently true. Many versions of the cosmological argument — such as the argument from contingency — assert that there cannot be any brute contingent truths and thus a necessarily true proposition must explain all of the contingent propositions (but then, of course, Inwagen’s problem comes about).

But the problem is worse. One might suppose that while the argument from contingency, etc, was false, some other argument for theism was true (presumably, Inwagen believes this to be the case). Consider any version of theism in which God is a necessarily existent being. There are two possibilities: 1. God had reason to create our universe or 2. God no reason to create our universe. Let’s examine each in turn.

If (1), then whatever God’s reasons were, must have originated in God’s essence. This follows from divine simplicity. But God’s essence, as a necessarily existent being, is necessarily existent. Thus, if God had reason to create the universe, then there are no contingent brute facts after all (how could there be?). The alternative is to deny God’s necessary existence (I have heard that Inwagen does deny that God is necessarily existent, and perhaps he has something like this in mind, but I have not read him on this point).

If (2), then God’s actions are arbitrary and random. Presumably, the theist would want to avoid this possibility (though, apparently, some of the medievals did not).

So I strongly suspect that the argument in this post is decisive against a number of conceptions of theism. Atheists, on the other hand, should be happy to hear that there is a good argument that it is necessary that there are brute, contingent facts.

Notice that the solution was to pose the existence of a proposition R which was brute & contingently true

That’s not quite right. R is necessarily contingent. A necessarily contingent proposition is such that, as a matter of metaphysical necessity, it cannot be explained. Failure to explain it does not display any shortcomings at all, either in omnipotence, omniscience or perfect goodness. So, the argument goes, PSR cannot reasonably require an explanation for such propositions. It cannot reasonably require that anyone, including God, do the impossible.

The point could be put another way. If PSR entails the impossible, then PSR is false. But if PSR entails that a perfectly rational being explains propositions that cannot be explained, then PSR entails the impossible. So, you could either reject PSR, or give it the right interpretation: it should be understood as requiring God or a perfectly rational being to explain every proposition that could be explained.

>>R is necessarily contingent. A necessarily contingent proposition is such that, as a matter of metaphysical necessity, it cannot be explained. Failure to explain it does not display any shortcomings at all, either in omnipotence, omniscience or perfect goodness.

Well, I understand that R is necessarily contingent and that this implies that a God is not needed to explain it (because it is metaphysically impossible that anything could explain it). What I don’t understand is what would distinguish a brute contingent fact from a necessarily contingent fact. By a brute contingent fact, I mean a fact that has no further explanation and which is contingent. As I understand it, that’s exactly what a necessarily contingent fact is.

Atheists can claim that the universe’s existence just is; i.e. that the universe’s existence is a brute contingent fact. This was essentially Russell’s reply to Copleston, if memory serves me correctly.

>>[PSR] should be understood as requiring God or a perfectly rational being to explain every proposition that could be explained.

Right, but, as I understood it, what is up for debate between the atheist and the theist is whether or not a perfectly rational being is required to explain the universe’s existence. Your solution, while saving the PSR, opens space for an atheist to claim that the universe’s existence is a necessarily contingent fact, having no explanation and therefore no need for a creator.

What I don’t understand is what would distinguish a brute contingent fact from a necessarily contingent fact. By a brute contingent fact, I mean a fact that has no further explanation and which is contingent. As I understand it, that’s exactly what a necessarily contingent fact is.

A brute contingent fact is not in general necessarily contingent: some are, some aren’t.. What corresponds to a necessarily contingent fact is a necessarily brute fact. The universe, as a conjunction of some contingent facts, would require an explanation, since it can be explained. But the largest contingent fact would not require an explanation, since it is necessarily contingent (and so necessarily brute).

There are some interesting questions here, though. The fact that R has no necessary explanation does not entail that P has no contingent explanation. Suppose R includes P + ◊P’. P cannot include the contingent fact G that God actualizes or verifies P, otherwise G would explain itself. But there is a possible, contingent explanation G for P. If P might have an explanation, then it has an explanation. Alternatively, R might include P + G + ◊P’. In that case, G explains P and nothing could explain R.

@Michael: Van Inwagen’s reductio of PSR relies on the premise that no contingent propositions explain themselves. But what about propositions that report Libertarian free will decisions? They seem both contingent and self-explaning. If so, Libertarians have a good reason to reject Van Inwagen’s reductio.

If libertarian freedom entails indeterminism, as it seems to, then libertarianism looks incompatible with PSR (given the PvI/Bennett argument). But I think (see above) that even strong PSR is compatible with the sort of contingency necessary for libertarianism. I actually think that God’s actions might be necessary, and he might necessarily create the world, and those even facts are consistent with widespread contingency. But it is a very long story.

Hi Michael, I would say that libertarianism does not look incompatible with PSR. If contingent truths that report libertarian free will decisions (e.g., “Michael freely decides to write a blogpost on PSR”) are self-explaining truths, then there seems to be a sufficient reason for these truths. That is to say, if we take it that self-explanation constitutes a sufficient reason (which seems plausibly true). Moreover, a Libertarian could hold that there is a sufficient reason for all other contingent truths as well, namely nomic causation (for stones rolling of a mountain, atoms decaying, etc.) and for some contingent truths perhaps a combination of nomic and free will causation. In the latter case the sufficient reason would be a cause that is partially nomic and partially based on a free will act.

The universe U is the created part of the world W, the part that requires explanation. Construed as a contingent conjunction, it would include just a subset of the true propositions in W. There are larger true contingent conjunctions in W.

I’m with you until the very end of this post and then I get confused. This is supposed to be a “way out this problem” that does not require weakening PSR or adopting a weaker notion of explanation. So, I was assuming, it’s supposed to allow us to hold on to PSR. How does it do that? The Principle of Sufficient Reason–as addressed in van Inwagen and Bennett’s arguments, I think–is fully general. It says that for any true proposition, p, there is another proposition, q, which explains p. If that’s the version of PSR you’re concerned with holding on to without having to weaken, I don’t see how you do it. Suppose you’re right that R “is a contingent proposition that is impossible to explain.” Then, isn’t R just a counter-example to the full-throated PSR? Maybe the idea is that R shows us that we should endorse a version of PSR that says that, for any proposition, p, where p could have an explanation, there is a proposition q such that q explains p. In that case, though, haven’t you weakened PSR (at least, as PvI was thinking of it)? Also, doesn’t PvI’s argument still work against that version of PSR, since P (the conjunction of all contingently true propositions) is a proposition that could have an explanation?

I’m wondering why you think the quantifier in PSR is supposed to be read unrestrictedly. The principle itself doesn’t say that it is unrestricted. Moral principles also fail to state that they are restricted to actions that we are able to perform. So, when I’m urged to maximize overall utility, I do not read that as unrestrictedly requiring that I perform the maximizing action. It might be that I cannot do that; it is surely restricted to the maximizing actions I can perform. I also don’t take that as weakening the principle. I’m simply giving the principle a chance to be true

I do not read PSR unrestrictedly, either, since I’d like to give that principle a chance of being true. I take the principle as requiring that everything that can be explained be explained. If I read the principle as stating that everything, even propositions that cannot be explained, must have an explanation, then PSR would be a non-starter.

Fair enough. I guess I would have thought that if you were going to carefully state a principle about maximing overall utility, you would explicitly limit that principle to the actions that you can then perform. You would say something like “for any action that is then open to you, take the one that maximizes utility.” van Inwagen is pretty careful about the way he states PSR. So, I would have thought that he would make explicit any restriction on the quantifier he intended. I guess I don’t think much hinges on whether you’re proposing a slightly weakened PSR or not.

Can you say more about about two things?

First, doesn’t PvI’s argument still threaten PSR as you’re understanding it? R can’t be an explanation for P because it has P as a conjunct. Are you thinking that the existence of R shows that there are contingent propositions that cannot be explained and, as a result, there might be some other (unnamed) proposition which is contingently true, can’t be explained, but is the explanation for P?

Second, can you say more about why you think that R is necessarily contingent?

Hi Michael, I agree with you that contingent truths reporting l(ibertarian)-free *actions* are not self-explanatory. After all, these truths are explained by contingent truths reporting l-free *decisions*. But contingent truths reporting l-free *decisions* surely seem at least prima facie self-explanatory. It is not just me that holds this. For example Gale and Pruss write: “[A]n indeterministic [decision] is a self-explainer only if it is a free [decision]” (A New Cosmological Argument, 1999, p. 472). One could even argue that it is definitory for libertarianism that contingent truths reporting l-free decisions are self-explanatory. So, if you believe that these truths are not self-explanatory, then it seems to me that you need to provide an argument for that.

van Inwagen did not state the the principle applies unrestrictedly, probably because he did not consider the possibility of a necessarily contingent proposition. That’s my guess.

Let me take the second question first. I think I said that R = P + ◊P’, where P’ is the conjunction of propositions such that P’ is true only if W’ obtains; P on the other hand is a conjunction of propositions which is true only if W obtains. The simple idea is that P’ is the book on W’ and P the book on W. Suppose the big conjunct R has an explanation Q such that ☐(Q ⊃ R) & ☐Q. In that case we have ☐R. But if ☐R, then by the same closure principle just used, we derive ☐P. (R entails P and R is necessary, so P is necessary). But ☐P entails that world W is necessary: there are, then, no other possible worlds. That is inconsistent with ◊P’, which is also entailed by R, and entails that W’ is possible (where W’ ≠ W). So, if P’ is possible, it can’t be the case that P is necessary. But then R is necessarily contingent.

Now to the first question. If R is necessarily contingent, then P has, at most, a contingent explanation. But that’s what we want. P too turns out to be necessarily contingent; R entails that. I’ll leave that as an exercise for the reader (assuming someone is still reading)

But this is just an assertion. If Pruss also asserts it, it’s still just an assertion. Where’s the argument that it is self-explanatory? There are some obvious reasons to think that it is not self-explanatory. Ask what Smith did. Suppose the answer is that he consumed some arsenic that caused his death. I say “explain why he did that”. You say: “because he decided to”. Certainly someone might insist on an answer to the question, why did he decide to do that?! Asserting that such a decision is self-explanatory won’t cut any ice. If there’s no further explanation, then his desiring to do that is a brute fact, not self-explanatory. It might be all the libertarian can say, but that’s the libertarian’s problem.

No, that is not all the libertarian can say. On libertarianism an agent may have reasons for and reasons against his or her free decision. But, and that is what libertarianism is all about, these reasons together do not constitute a *sufficient* explanation for the agent’s free decision. On libertarianism, the only sufficient explanation for a free decision is the fact that the agent in question, taking all reasons pro and con into account, freely chose to decide in that way. Thus the decision, if it is a libertarian free decision, is self-explanatory. Now, you may find libertarianism itself implausible or even indefensible. Fine. But then you need to do more than merely assert that.

Thus the decision, if it is a libertarian free decision, is self-explanatory.

What do you mean ‘thus’? That is not an argument whose conclusion is that the decision is self-explanatory. A decision whose reasons are insufficient for an agent to take that decision is not self-explanatory (not by a long shot), and it is not even explained by the reasons you adduce. What you are describing is a brute fact.

“Given all the reasons you have available, could you have decided to do ~A rather than A?” If the answer is “yes”, and if the agent decided to do A in spite of lacking sufficient reason to do so, then the agent made his decision in the absence of sufficient reason to do it. I have no idea how you arrive at the conclusion that it is “thus” self-explanatory: the reasons that are missing would be the explanation that is missing. The deciding itself is not a reason and so is not itself an explanation of anything. Precisely what is lacking in the decision is an explanation, whether a self-explanation or otherwise.

I agree that all that is left for the libertarian is the simple fact that the agent decided. But you really ought to stop calling that “thus” self-explanatory. What it is thus is a brute fact, lacking explanation. For all of the reasons and explanations available, all agree, he might well have decided otherwise. It is just the role of sheer chance in the process that he didn’t.

1. Isn’t R one of the conjuncts of P, assuming here just for the sake of the argument that there are such conjunctions (why not challenge the coherence of the conjunction)?

In fact, if P is contingent and true, and P is the the conjunction of all propositions that are contingent and true, then it seems P is a conjunct of itself. Moreover, if N is a necessarily true proposition, then P&N is also contingent and true. But ◊P’ seems to be a necessary truth, so P&◊P’ would seem to be a contingent truth, and thus a conjunct of P.
Of course, if Van Inwagen’s argument is correct, then it seems P is not contingent after all…

2. Why shouldn’t a defender of Van Inwagen’s argument reply by just making the same argument?

Let’s R be the conjunction of P&◊P’. Then, by PSR, there is a necessary explanation Q’.
So, ☐(Q’ ⊃ R) & ☐Q’
From that, it follows that ☐R. From that, it follows that ☐P. It follows that W’=W, P’=P, and R is P&◊P.

@Michael: Your “sheer chance”-response amounts to the well-known randomness objection against libertarianism. You now starting to argue against libertarianism itself accords with the last three sentences of my previous post. So good. Nevertheless I trust that you agree that this is not the place to launch an in-depth discussion on the tenability of libertarianism. It would become, to use your words, a long störy. As to the question I was focussing on, namely the question of whether on libertarianism reasons-informed (not reasons-determined) free decisions are self-explaining (i.e., sufficiently explained by the fact that the agent in question excercises her free will), see Alexander Pruss’ excellent book on PSR (especially chapter 7) if you have not already done so. For now, thanks for the discussion.

Well, we started out with this claim, according to which libertarianism is said to be compatible with PSR and libertarian decisions are both self-explaining and possess a sufficient reason.

… I would say that libertarianism does not look incompatible with PSR. If contingent truths that report libertarian free will decisions (e.g., “Michael freely decides to write a blogpost on PSR”) are self-explaining truths, then there seems to be a sufficient reason for these truths

So, I’m pretty sure I didn’t raise the issue of whether libertarianism itself was part of the problem, as you suggest I misleadingly did here.

You now starting to argue against libertarianism itself accords with the last three sentences of my previous post

I actually have no interest in launching a in-depth discussion of libertarianism, and I don’t think I said I did. But now you say you’re really focusing on this,

I was focussing on … the question of whether on libertarianism reasons-informed (not reasons-determined) free decisions are self-explaining (i.e., sufficiently explained by the fact that the agent in question exercises her free will

But now, we’re back to whether we’re talking about actions or not. Taking a decision, if that is what you mean by a ‘free decision’ is an action. But you denied above that you’re talking about actions. You say,

, I agree with you that contingent truths reporting l(ibertarian)-free *actions* are not self-explanatory. After all, these truths are explained by contingent truths reporting l-free *decisions*. But contingent truths reporting l-free *decisions* surely seem at least prima facie self-explanatory

So, free decisions are not actions, presumably, given your focus in this passage. But then how can a ‘free decision’ be sufficiently explained by the exercise of free will, if ‘free decisions’ are not actions? I assume it’s clear why this is all pretty difficult to follow.

Here’s what I would say. If a decision is self-explanatory, then it must be it’s own reason for occurring. It must be true that the reason Jones’ decision occurred is that Jone’s decision occurred, or the reason Smith decided to A is that Smith decided to A. But I think most everyone would agree that these are not reasons at all for those decisions. The very decision in these cases has to be the reason for the decision, but no one who is not forced into saying such things would say them. I can see why a libertarian would find himself having to say things like that. But it is prima facie (secunda facie, too) untrue that a decision D is an explanation for D. The absence of any reason R independent of D for the occurrence of, or the taking of, D is just what it means to say that the occurrence of D (the having of D, if you like) is a brute fact.

Now, I don’t deny that there might be other reasons R outside of D itself that might explain D. But this is no longer a question of D’s self-explanation. Given libertarianism, there might be reasons R that offer some stochastic explanation for D, for instance. I would not call them explanations that meet the standards of those who endorse PSR, but that’s another question.

R cannot be a conjunct of P, if van Inwagen’s argument is to succeed. Recall R = P + ◊P’, where P is the book on W and P’ is the book on W’ and (W ≠ W’). I derive that there is such an R from the fact (van Inwagen’s assumption) that P is contingent. If P is contingent, then there is an R fitting the description above. But such an R cannot have an explanation, since ☐R entails a contradiction.

Of course, in van Inwagen’s argument, we assume P is contingent for reductio. But, nonetheless, we still get R. And R entails that all other conjunctions of contingent propositions Z are necessarily contingent. So, the problem goes away.

I don’t see a problem with P including every contingent conjunction, if it is an infinite conjunction.

It follows from the definitions of P and P’ that R is a conjunct of P, just as P is a conjunct of P. That’s because any contingently true proposition is a conjunct of P, and P and R are contingently true proposition – it follows from the assumptions.

Regardless, if P is assumed to be contingent for reductio, essentially the same reductio might be deployed against R, it seems to me (that is the second point I raised above).

As for any problem for the conjunction P, I think that depends on whether propositions have subformulas, and some related matters. But that’s a different argument.

@Michael: A quick response (although, as the last sentence of my previous post suggests, I have in fact no interest to continue the discussion). When I wrote “sufficiently explained by the fact that the agent in question exercises her free will” I surely did not intend to change from decisions to actions. I intended to say “sufficiently explained by the fact that the agent in question does freely decide that way”. Now, you might be right that “exercises her free will” is not an appropriate way of intending to say “does freely decide that way”. But, since I *explicitly* pointed out earlier that I was talking about free decisions and not about free actions (and since I also *consequently* talked about decisions and not about actions in all my previous responses), it seems to me that me using once the (perhaps inappropriate) “exercises her fee will” should not have led you to think that my whole position on whether we talk about decisions or actions was suddenly changed. Further, I do start to get more and more the impression that you are not sufficiently aware of chapter 7 of Pruss’ book on PSR. If so, then I think it would be really helpful for you to have at least a look at it. Thanks again for the discussion.

I’ve of course read Pruss. But I don’t regard that as gospel, nor as especially persuasive. If you have an argument for your view other than ‘read Pruss’, I’d like to see it. If decisions are no longer actions on your view, but now other things, then I’m happy to discuss it. But if you don’t have an argument for it, I’m happy to move on.

I think van Inwagen intended P to be a conjunction all of whose conjuncts are true contingent propositions. It’s hard to tell, since the argument is not formally stated. In any case, I’ve been reading him that way and, I think, so has everyone else. But you’re right that I might have been clearer about that. Your second reductio does not entail (not that I can see, anyway) that R = P + ◊P. Recall that I introduced R as R = P + ◊P’, not some other proposition. Your attempted reductio entails rather that P + ◊P’ has an explanation, ☐Q. But ☐Q entails ☐R, and ☐R entails a contradiction, so we have to deny something in the argument. My candidate is the rejection of the claim that R has an explanation, since that leads directly to the contradiction. Can we seriously deny that R is a true contingent proposition? We can do so only if we deny that P is a true contingent proposition, since P entails R. But P is true by hypothesis! The only candidate to reject is that version of PSR that applies to necessarily contingent propositions.

The parallel I see is as follows.
van Inwagen’s argument (according to your OP) seems to be:

Let’s suppose there are contingently true propositions, and PSR is true.
Let P be the conjunction of all contingently true propositions. Then, P is contingent and true.
Then, there is some Q that explains P. But Q is not contingent, because if it were, then Q would be a contingently true proposition that explains itself, and that’s allegedly impossible.
So, we get that for some Q, ☐(Q ⊃ P) & ☐Q.
So, ☐P, contradicting the assumption that P is contingent.

Parallel:
Let’s suppose there are contingently true propositions, and PSR is true.
Let P be the conjunction of all contingently true propositions. Then, P is contingent and true.
Let P’ be the conjunction of all [contingently, I think; if P’ also includes necessary conjuncts, that is a slight difference but the parallel still holds] true propositions in possible world W’, which is different from the actual world W (by assumption, there is such a world).
Let R be P&◊P’. From the following assumptions, it follows that R is contingently true. Hence, there is some G that explains R. but then, G can’t be contingently true, because if it were, then it would be a conjunct of P, and thus R (which has P as a conjunct) would explain itself.
Hence, there is some necessary proposition Q such that ☐(Q ⊃ R) & ☐Q.
But then, R is necessary. Hence, P is necessary (because P is entaled by R), contradicting the assumption. Also, hence there is no possible world W’ =/= W, and so P’=P (of course, once you get a contradiction, you can get anything from there).

Granted, you may deny that PSR applies to R. But for that matter, you may as also deny that it applies to P (and in the end you do), so I’m not sure what difference R makes.
Also, granted, you may argue – as you do – that ☐R entails a contradiction, so we have to deny something in the argument. But then again, one may similarly reply to the original argument by pointing out that ☐P also entails a contradiction, so one needs to deny something in the argument.

At any rate, your stance denies that the PSR applies not only to R, but to P as well, because if PSR applied to P, then it would follow (by the reasoning above, and as before assuming no contingent proposition explains itself) that P is necessary, contradicting the assumption.

From a different angle, if R is necessarily contingent, it seems so is P (and it’s difficult to see what proposition could be possibly but not necessarily contingent).

What you seem to do is assume R is contingent and then prove not that R is necessary but that some other proposition R’ (such that R’ ≠ R) is necessary. You don’t prove anything about R that I can tell. We know that there is such an R (where R = P + ◊P’, where P ≠ P’), since P entails that there is such an R, and defenders of van Inwagen’s argument insist that there is such a P. Of course, there is an R’ too, where R’ = P + ◊P’, where P = P’. But R’ is not R. The main problem with taking the view that R can be explained is that it entails that some contradictions are true.

Another way out, which attracts me, is to deny that there can be conjunctions that big. For a conjunction of all contingent props is one of its own conjuncts, but no proposition can be one of its own conjunct. Also, it seems ad hoc to suppose that there is a conjunction of all contingent props unless in general, for any contingent ps, there is a conjunction of them. But this principle of conjunction falls prey to a Russellian contradiction: the conjunction of the ps that aren’t conjuncts of themselves is a conjunction of itself iff it isn’t.

This way out allows us to keep the fully unrestricted PSR.

(If we instead run the objection to PSR in terms of a proposition that *entails* all contingent ones, then the circularity problem goes away. Compare: the axioms of geometry explain the Pythagorean theorem, even though the Pythagorean theorem entails those axioms–and vice versa.)

Thanks for this! You write,Another way out, which attracts me, is to deny that there can be conjunctions that big. For a conjunction of all contingent props is one of its own conjuncts
Yes, Angra alluded to this problem above. I’m not sure it would worry PvI’s argument. For, if you take the smallest conjunction of distinct atomic contingent propositions P, P will entail all of the larger conjunctions C formed from it’s conjuncts. For instance, if A1 & A2 = P, then P will entail (A1 v A2) & A2, and P will entail (A1 & A2) & (A2 & A1), etc. So, he needs only the smallest conjunction.

I’ll have ot think about the Russellian objection, which I don’t see offhand.

“PSR requires that there is some explanation q such that q explains p if and only if ☐(q ⊃ p)”

One reasonable take on the PSR (the one in my PSR book) is: For all contingent true p, p has an explanation. There is no requirement that the explanation entails p.

For the record, Van Inwagen used to hold that if q explains p, then q entails p, but I believe I convinced him otherwise at one of the St Thomas summer seminars. (The argument that I used was, roughly, that facts about laws can explain physical events even if the possibility of a miracle makes the explanation be non-entailing.) And indeed the philosophy of science literature on explanation by and large accepts that there are non-entailing explanations (e.g., stochastic ones).

Van Inwagen still holds that if q explains p, and q is necessary, then p is necessary. (But he doesn’t any longer derive this from the thesis that if q explains p, then q entails p.)

Josh:

You probably still get a version of the van Inwagen problem if you consider the conjunction of all contingent *fundamental* facts (though van Inwagen won’t want to run the argument this way, since he doesn’t like the notion of fundamentality), or restrict the conjunction in some other reasonable way.

There are two separate issues here. The first is, as a response to the van Inwagen/Bennett argument, the rejection of the notion of explanation in the argument is too global. It is in any case, too global for me to find it interesting. The second concerns whether it might be true that a perfectly rational being has a better reason R to actualize world W than to actualize any other world W’ (even if R does not obviously entail W) and it be possible that he actualize W’. I don’t find it credible that a perfectly rational being could actualize any such W’. So, I’m not unhappy with the van Inwagen/Bennett notion of explanation as applied to perfectly rational beings. I realize that they do not explicitly refer to perfect beings in the formulation of their argument.

Let me just observe quickly the odd fact that if R is a reason to actualize W and R’ a reason to actualize W’, R is just slightly better than R’, and W and W’ exhaust the worlds, then, as I said above, R does not obviously entail W (if I can speak loosely of entailment of a world here). R does entail W, just not obviously. I see no way to avoid this other than to say that a perfectly rational being *might* act contrary to his better reasons (as by hypothesis he has for W).